The man pages for cacos(), cacosh(), catan(), catanh()
contain incorrect formulae describing the functions.
As reported by Richard B. Kreckel:
cacos, cacosf, cacosl:
The formula given in the man page
cacos(z) = -i clog(z + csqrt(z * z - 1))
gives wrong results in second and fourth quadrant of
complex plain.
The formula
cacos(z) = -i clog(z + I*csqrt(1 - z * z))
gives correct results.
catan, catanf, catanl:
The formula given in the man page
catan(z) = 1 / 2i clog((1 + iz) / (1 - iz))
gives wrong results on the negative imaginary axis beginning
at -I (along one of the two branch cuts). Besides, the formula
is written in an ambiguous way.
The formula
catan(z) = (clog(1 + iz) - clog(1 - iz)) / 2i
gives correct results.
cacosh, cacoshf, cacoshl:
The formula given in the man page
cacosh(z) = (0.5) * clog((1 + z) / (1 - z))
gives wrong results everywhere in the complex plain.
(The formula seems to be copied from the one for catanh,
where it is sometimes correct.)
The formula
cacosh(z) = 2 * clog(csqrt((z + 1)/2) + csqrt((z - 1)/2))
gives correct results.
catanh, catanhf, catanhl:
The formula given in the man page
catanh(z) = 0.5 * clog((1 + z) / (1 - z))
gives wrong results on the positive real axis beginning at 1
(along one of the two branch cuts).
The formula
catanh(z) = 0.5 * (clog(1 + z) - clog(1 - z))
gives correct results.
I've also checked casin, casinf, casinl, casinh, casinhf, and
casinhl and the formulae given there
casin(z) = -i clog(iz + csqrt(1 - z * z))
casinh(z) = clog(z + csqrt(z * z + 1))
are actually correct.
Reported-by: Richard B. Kreckel <kreckel@ginac.de>
Reviewed-by: Andries Brouwer <Andries.Brouwer@cwi.nl>
Signed-off-by: Michael Kerrisk <mtk.manpages@gmail.com>